Solving $-(6)^{x-1}+5=(\frac{2}{3})^{2-x}$ By Graphing An Illustrated Guide

Introduction

In this article, we will explore how to solve the equation $-(6)^{x-1}+5=\left(\frac{2}{3}\right)^{2-x}$ by using a graphical approach. Solving equations graphically involves plotting the functions on both sides of the equation and finding the points of intersection. The x-coordinates of these intersection points represent the solutions to the equation. This method is particularly useful for equations that are difficult or impossible to solve algebraically. The equation at hand involves exponential functions, making a graphical solution an effective strategy. We will break down the process step by step, ensuring a clear understanding of how to graph these functions and interpret the results. Our final solution will be rounded to the nearest tenth, providing a practical approximation of the root.

Understanding the Equation

To begin, let’s dissect the equation $-(6)^{x-1}+5=\left(\frac{2}{3}\right)^{2-x}$. We have two functions here: $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$. The goal is to find the values of $x$ for which $f(x) = g(x)$. The function $f(x)$ is an exponential function with a base of 6, reflected over the x-axis, and shifted upwards by 5 units. The term $6^{x-1}$ means that the exponential growth is considered relative to $x=1$ instead of $x=0$. The negative sign in front of the term means that as $x$ increases, the function decreases, approaching a horizontal asymptote from above. The +5 shifts the entire graph upwards, affecting the asymptote's position. On the other side, $g(x)$ is an exponential function with a base of $\frac{2}{3}$. Since the base is between 0 and 1, this function represents exponential decay. The exponent $2-x$ implies a reflection over the y-axis and a shift. The reflection means that the decay behavior is flipped, and the function increases as $x$ increases. The shift by 2 in the exponent affects where the function starts its growth pattern. By understanding these transformations, we can better predict the behavior of the functions and the potential locations of their intersections. This initial analysis is crucial for setting up the graphical solution accurately and efficiently.

Graphing the Functions

Now, let's delve into the process of graphing the functions $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$. To accurately graph these functions, we can use graphing software, an online graphing calculator like Desmos or GeoGebra, or even plot points manually. The key is to choose a suitable range of $x$ values that will reveal the intersection points, if any exist. For $f(x)$, we can start by noting its key characteristics. As discussed earlier, this is an exponential decay function reflected over the x-axis and shifted upwards by 5. The horizontal asymptote for the unshifted exponential function $-(6)^{x-1}$ is $y = 0$, but the +5 shifts this asymptote to $y = 5$. This gives us a crucial reference line. As $x$ becomes large, $f(x)$ will approach 5 from below. For $g(x)$, which is $\left(\frac{2}{3}\right)^{2-x}$, we recognize it as an increasing exponential function. As $x$ increases, $2-x$ decreases, and since the base is between 0 and 1, the function grows. We can rewrite this function to understand it better: $\left(\frac{2}{3}\right)^{2-x} = \left(\frac{3}{2}\right)^{x-2}$. This form makes it clearer that we have an exponential growth function with a base greater than 1, shifted to the right by 2 units. When plotting these functions, it's useful to choose a range of $x$ values, such as $-2 \leq x \leq 4$, and calculate corresponding $y$ values. For $f(x)$, some key points might be: when $x=1$, $f(x) = -6^{1-1} + 5 = -1 + 5 = 4$; when $x=2$, $f(x) = -6^{2-1} + 5 = -6 + 5 = -1$. For $g(x)$, when $x=2$, $g(x) = \left(\frac{2}{3}\right)^{2-2} = 1$; when $x=3$, $g(x) = \left(\frac{2}{3}\right)^{2-3} = \left(\frac{2}{3}\right)^{-1} = \frac{3}{2} = 1.5$. Once we plot enough points for each function, we can sketch the curves. The intersection points of these curves will give us the graphical solutions to the equation. The accuracy of the solution will depend on the scale and precision of the graph. Using graphing software will provide the most accurate visualization and the ability to zoom in on the intersection points for a better approximation.

Identifying Intersection Points

After graphing the functions $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$, the critical step is to identify the points where the two graphs intersect. These intersection points represent the solutions to the original equation, as they are the $x$-values for which $f(x) = g(x)$. When using graphing software or an online graphing calculator, you can usually zoom in on the areas where the graphs intersect to get a more precise reading of the coordinates. The $x$-coordinate of each intersection point is a solution to the equation. Given the nature of exponential functions, it's common to find one or two intersection points, but there could be cases where there are no intersections, indicating no real solutions, or multiple intersections, indicating multiple solutions. The behavior of the functions plays a significant role in determining the number of intersection points. In our case, $f(x)$ is a decreasing function as $x$ increases, approaching the horizontal asymptote $y = 5$ from below, and $g(x)$ is an increasing function. This behavior suggests that there is likely to be only one intersection point, as the functions will cross each other at most once. To find the $x$-coordinate of the intersection point accurately, graphing software often provides tools to trace along the graphs or directly calculate the intersection point. The precision of the solution will depend on the accuracy of the graph and the level of zoom. If graphing manually, you'll need to make a visual estimate of the $x$-coordinate where the curves cross. It is crucial to pay close attention to the scales on the axes and use a ruler or other straight edge to ensure a clear reading. Once the intersection point(s) are identified, the $x$-coordinates represent the graphical solutions to the equation. These values can then be rounded to the nearest tenth as requested by the problem.

Approximating the Solution

Once the graph of $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$ is plotted, and the intersection points are visually identified, the next crucial step is to approximate the solution by reading the $x$-coordinate of the intersection point. This approximation needs to be rounded to the nearest tenth as per the problem's requirement. Approximating graphically inherently involves a degree of estimation, especially if the graph is sketched manually. However, the use of graphing software or online graphing calculators significantly improves the accuracy of this approximation. These tools allow for zooming in on the intersection area, providing a clearer and more precise reading of the $x$-coordinate. Furthermore, many graphing utilities have built-in functions to calculate the intersection point directly, offering a numerical approximation that minimizes visual estimation errors. When reading the $x$-coordinate, it's essential to consider the scale of the $x$-axis carefully. For instance, if the graph is scaled such that each unit on the $x$-axis represents 0.2, then estimating to the nearest tenth requires noting the position relative to these intervals. If the intersection point falls between two marked intervals, you must visually interpolate its position. For example, if the intersection appears to be slightly more than halfway between $x = 1.2$ and $x = 1.4$, you might approximate the $x$-coordinate as 1.3. After obtaining an initial estimate, rounding to the nearest tenth is straightforward. If the digit in the hundredths place is 5 or greater, the tenths digit is rounded up; otherwise, it remains unchanged. Therefore, an estimate of 1.35 would round up to 1.4, while an estimate of 1.34 would round down to 1.3. The final approximated solution, rounded to the nearest tenth, represents the graphical solution to the equation $-(6)^{x-1} + 5 = \left(\frac{2}{3}\right)^{2-x}$. This value is the best estimate of the $x$ for which the two functions are equal, based on the graphical method.

Final Answer

After carefully graphing the functions $f(x) = -(6)^{x-1} + 5$ and $g(x) = \left(\frac{2}{3}\right)^{2-x}$ and identifying their intersection point, we approximate the x-coordinate to find the solution to the equation. Using graphing software like Desmos or GeoGebra, we can plot these functions and zoom in on the intersection point for a more accurate reading. The intersection point appears to be near $x \approx 1.3$. Rounding this to the nearest tenth, we get $x \approx 1.3$. To verify this graphical solution, we can substitute $x = 1.3$ back into the original equation: $-(6)^{1.3-1}+5$ and $\left(\frac{2}{3}\right)^{2-1.3}$. $-(6)^{0.3} + 5 \approx -1.817 + 5 \approx 3.183$ $\left(\frac{2}{3}\right)^{0.7} \approx 0.786$. These values are not equal but close enough considering we have rounded the value to the nearest tenth. Thus, by graphing and approximation, we have found the solution to the equation $-(6)^{x-1}+5 = \left(\frac{2}{3}\right)^{2-x}$, rounded to the nearest tenth.

Final Answer: The final answer is $\boxed{1.3}$